1.2 A Brief History of Diffusion Science

The tortuous history of the development of today’s understanding of the above phenomena and other aspects of the theory of diffusion of gases in porous solids is described in detail by Cunningham and Williams (1980). It is a fascinating story that chronicles the major developments in the science of diffusion and analyzes in some depth the many mistakes and misconceptions that occurred along the way. The story begins with Thomas Graham (1833), who observed and reported with little explanation all of the phenomena described in the above discussion. Graham also conducted experiments on diffusion in liquids, the results of which were an important contribution to the development of the first constitutive equation for diffusion, now known as Fick’s law (Fick, 1855).

For the next 100 years, Fick’s law was commonly thought to be the only constitutive equation necessary to codify the results of diffusion experiments. Fick’s law, developed for diffusion in liquids but often assumed to apply to gases as well, predicts that the components of an isobaric binary gas diffuse at equal rates in opposite directions and that no diffusion-engendered bulk flow occurs. While this prediction was inconsistent with Graham’s results, it was apparently correct in other experiments performed under different working conditions. Consequently, Graham’s results came to be discredited and were generally ignored or forgotten until the middle of the twentieth century when they were resurrected by Hoogschagen (1955). The simple empirical equation that conveniently expresses the salient result of Graham’s experiments was eventually elevated to the status of a law on par with Fick’s law and is now known as Graham’s law of diffusion.

An important advance in the theoretical study of diffusion in gases occurred during the second half of the nineteenth century. According to Cussler (1997), Clerk Maxwell recognized as early as 1860 that diffusion generates bulk flow. He modified Fick’s law to include advection by diffusion-engendered bulk flow and interpreted the equation in terms of the rate of momentum loss of a component due to molecular collisions of that component with molecules of the second species (Cunningham and Williams, 1980). Hoogschagen (1955), who was evidently unaware of Graham’s work, conducted independent experiments that showed the mole fluxes of the individual components were related to each other in the manner discovered by Graham. He combined what is now known as Graham’s law with Maxwell’s constitutive equation to derive the first correct expression for steady diffusion in an isobaric, binary gas occupying the voids of a porous medium.

Hoogschagen’s expression for isobaric diffusion applies when the resistance to diffusion is dominated by the collisions between the molecules of one component with those of the other. This is the situation that prevails in the so-called molecular regime. It was yet another several years before the theory of diffusion in gases in porous media was extended by E. A. Mason and coworkers (Evans III et al., 1961, 1962; Mason et al., 1967; Mason and Malinauskas, 1983) to include the resistance to diffusion that results from collisions of component molecules with solid particles embedded in the diffusion path. This prodigious work culminated in what is now known as the Dusty Gas Model, in which solid particles embedded in the system are regarded as giant immobile molecules. The Dusty Gas Model includes rigorously derived constitutive equations for multi-component transport in porous solids, together with physical interpretations and predictive expressions for the transport coefficients that emerge. The Dusty Gas Model constitutive equation for diffusion reduces to the familiar Fick’s law only under very restrictive conditions.


Flux Equations for Gas Diffusion in Porous Media Copyright © 2021 by David B. McWhorter. All Rights Reserved.